Found problems: 85335
2025 Vietnam National Olympiad, 5
Consider a $3k \times 3k$ square grid (where $k$ is a positive integer), the cells in the grid are coordinated in terms of columns and rows: Cell $(i, j)$ is at the $i^{\text{th}}$ column from left to right and the $j^{\text{th}}$ row from bottom up. We want to place $4k$ marbles in the cells of the grid, with each cell containing at most one marble, such that
- Each row and each column has at least one marble
- For each marble, there is another marble placed on the same row or column with that marble.
a) Assume $k=1$. Determine the number of ways to place the marbles to satisfy the above conditions (Two ways to place marbles are different if there is a cell $(i, j)$ having a marble placed in one way but not in the other way).
b) Assume $k \geq 1$. Find the largest positive integer $N$ such that if we mark any $N$ cells on the board, there is always a way to place $4k$ marbles satisfying the above conditions such that none of the marbles are placed on any of the marked cells.
2008 Mediterranean Mathematics Olympiad, 2
Determine whether there exist two infinite point sequences $ A_1,A_2,\ldots$ and $ B_1,B_2,\ldots$ in the plane, such that for all $i,j,k$ with $ 1\le i < j < k$,
(i) $ B_k$ is on the line that passes through $ A_i$ and $ A_j$ if and only if $ k=i+j$.
(ii) $ A_k$ is on the line that passes through $ B_i$ and $ B_j$ if and only if $ k=i+j$.
[i](Proposed by Gerhard Woeginger, Austria)[/i]
2011 Korea Junior Math Olympiad, 3
Let $x, y$ be positive integers such that $gcd(x, y) = 1$ and $x + 3y^2$ is a perfect square. Prove that $x^2 + 9y^4$ can't be a perfect square.
VII Soros Olympiad 2000 - 01, 11.7
Consider all possible functions defined for $x = 1, 2, ..., M$ and taking values $​​y = 1, 2, ..., n$. We denote the set of such functions by $T.$ By $T_0$ we denote the subset of $T$ consisting of functions whose value changes exactly by $ 1$ (in one direction or another) when the argument changes by $1$. Prove that if $M\ge 2n-4$, then among the functions from of the set $T$, there is a function that coincides at least at one point with any function from $T_0$. Specify at least one such function. Prove that if $M <2n-4$, then there is no such function.
2001 India IMO Training Camp, 3
Points $B = B_1 , B_2, \cdots , B_n , B_{n+1} = C$ are chosen on side $BC$ of a triangle $ABC$ in that order. Let $r_j$ be the inradius of triangle $AB_jB_{j+1}$ for $j = 1, \cdots, n$ , and $r$ be the inradius of $\triangle ABC$. Show that there is a constant $\lambda$ independent of $n$ such that :
\[(\lambda -r_1)(\lambda -r_2)\cdots (\lambda -r_n) =\lambda^{n-1}(\lambda -r)\]
2022 Iranian Geometry Olympiad, 2
Two circles $\omega_1$ and $\omega_2$ with equal radius intersect at two points $E$ and $X$. Arbitrary points $C, D$ lie on $\omega_1, \omega_2$. Parallel lines to $XC, XD$ from $E$ intersect $\omega_2, \omega_1$ at $A, B$, respectively. Suppose that $CD$ intersect $\omega_1, \omega_2$ again at $P, Q$, respectively. Prove that $ABPQ$ is cyclic.
[i]Proposed by Ali Zamani[/i]
2008 Cono Sur Olympiad, 4
What is the largest number of cells that can be colored in a $7\times7$ table in such a way that any $2\times2$ subtable has at most 2 colored cells?
2023 Brazil National Olympiad, 4
Let $x, y, z$ be three real distinct numbers such that
$$\begin{cases} x^2-x=yz \\ y^2-y=zx \\ z^2-z=xy \end{cases}$$ Show that $-\frac{1}{3} < x,y,z < 1$.
1997 Tournament Of Towns, (527) 4
A square is cut into 25 smaller squares, exactly 24 of which are unit squares. Find the area of the original square.
(V Proizvolov)
2012 ELMO Shortlist, 7
Let $\triangle ABC$ be an acute triangle with circumcenter $O$ such that $AB<AC$, let $Q$ be the intersection of the external bisector of $\angle A$ with $BC$, and let $P$ be a point in the interior of $\triangle ABC$ such that $\triangle BPA$ is similar to $\triangle APC$. Show that $\angle QPA + \angle OQB = 90^{\circ}$.
[i]Alex Zhu.[/i]
2010 Saint Petersburg Mathematical Olympiad, 6
For positive is true $$\frac{3}{abc} \geq a+b+c$$
Prove $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \geq a+b+c$$
2017 Ukraine Team Selection Test, 5
Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.
2009 QEDMO 6th, 11
The inscribed circle of a triangle $ABC$ has the center $O$ and touches the triangle sides $BC, CA$ and $AB$ at points $X, Y$ and $Z$, respectively. The parallels to the straight lines $ZX, XY$ and $YZ$ the straight lines $BC, CA$ and $AB$ (in this order!) intersect through the point $O$. Points $K, L$ and $M$. Then the parallels to the straight lines $CA, AB$ and $BC$ intersect through the points $K, L$ and $M$ in one point.
2019 Simon Marais Mathematical Competition, A4
Suppose $x_1,x_2,x_3,\dotsc$ is a strictly decreasing sequence of positive real numbers such that the series $x_1+x_2+x_3+\cdots$ diverges.
Is it necessary true that the series $\sum_{n=2}^{\infty}{\min \left\{ x_n,\frac{1}{n\log (n)}\right\} }$ diverges?
2011 Iran Team Selection Test, 3
There are $n$ points on a circle ($n>1$). Define an "interval" as an arc of a circle such that it's start and finish are from those points. Consider a family of intervals $F$ such that for every element of $F$ like $A$ there is almost one other element of $F$ like $B$ such that $A \subseteq B$ (in this case we call $A$ is sub-interval of $B$). We call an interval maximal if it is not a sub-interval of any other interval. If $m$ is the number of maximal elements of $F$ and $a$ is number of non-maximal elements of $F,$ prove that $n\geq m+\frac a2.$
2018 Ramnicean Hope, 3
Prove that for any noncollinear points $ A,B,C $ and positive real numbers $ x,y, $ the following inequality is true.
$$ xAB^2- \frac{xy}{x+y}BC^2 +yCA^2\ge 0 $$
[i]Constantin Rusu[/i]
1998 Iran MO (3rd Round), 6
For any two nonnegative integers $n$ and $k$ satisfying $n\geq k$, we define the number $c(n,k)$ as follows:
- $c\left(n,0\right)=c\left(n,n\right)=1$ for all $n\geq 0$;
- $c\left(n+1,k\right)=2^{k}c\left(n,k\right)+c\left(n,k-1\right)$ for $n\geq k\geq 1$.
Prove that $c\left(n,k\right)=c\left(n,n-k\right)$ for all $n\geq k\geq 0$.
2022 Iran-Taiwan Friendly Math Competition, 1
Let $k\geqslant 2$ be an integer, and $a,b$ be real numbers. prove that $a-b$ is an integer divisible by $k$ if and only if for every positive integer $n$
$$\lfloor an \rfloor \equiv \lfloor bn \rfloor \ (mod \ k)$$
Proposed by Navid Safaei
2021 Saudi Arabia IMO TST, 12
Let $p$ be an odd prime, and put $N=\frac{1}{4} (p^3 -p) -1.$ The numbers $1,2, \dots, N$ are painted arbitrarily in two colors, red and blue. For any positive integer $n \leqslant N,$ denote $r(n)$ the fraction of integers $\{ 1,2, \dots, n \}$ that are red.
Prove that there exists a positive integer $a \in \{ 1,2, \dots, p-1\}$ such that $r(n) \neq a/p$ for all $n = 1,2, \dots , N.$
[I]Netherlands[/i]
2023 Pan-American Girls’ Mathematical Olympiad, 4
In an acute-angled triangle $ABC$, let $D$ be a point on the segment $BC$. Let $R$ and $S$ be the feet of the perpendiculars from $D$ to $AC$ and $AB$, respectively. The line $DR$ intersects the circumcircle of $BDS$ at $X$, with $X \neq D$. Similarly, the line $DS$ intersects the circumcircle of $CDR$ at $Y$, with $Y \neq D$. Prove that if $XY$ is parallel to $RS$, then $D$ is the midpoint of $BC$.
2018 Bundeswettbewerb Mathematik, 2
Find all real numbers $x$ satisfying the equation
\[\left\lfloor \frac{20}{x+18}\right\rfloor+\left\lfloor \frac{x+18}{20}\right\rfloor=1.\]
2021 Indonesia TST, C
A square board with a size of $2020 \times 2020$ is divided into $2020^2$ small squares of size $1 \times 1$. Each of these small squares will be coloured black or white. Determine the number of ways to colour the board such that for every $2\times 2$ square, which consists of $4$ small squares, contains $2$ black small squares and $2$ white small squares.
2005 Postal Coaching, 20
In the following, the point of intersection of two lines $ g$ and $ h$ will be abbreviated as $ g\cap h$.
Suppose $ ABC$ is a triangle in which $ \angle A \equal{} 90^{\circ}$ and $ \angle B > \angle C$. Let $ O$ be the circumcircle of the triangle $ ABC$. Let $ l_{A}$ and $ l_{B}$ be the tangents to the circle $ O$ at $ A$ and $ B$, respectively.
Let $ BC \cap l_{A} \equal{} S$ and $ AC \cap l_{B} \equal{} D$. Furthermore, let $ AB \cap DS \equal{} E$, and let $ CE \cap l_{A} \equal{} T$. Denote by $ P$ the foot of the perpendicular from $ E$ on $ l_{A}$. Denote by $ Q$ the point of intersection of the line $ CP$ with the circle $ O$ (different from $ C$). Denote by $ R$ be the point of intersection of the line $ QT$ with the circle $ O$ (different from $ Q$). Finally, define $ U \equal{} BR \cap l_{A}$. Prove that
\[ \frac {SU \cdot SP}{TU \cdot TP} \equal{} \frac {SA^{2}}{TA^{2}}.
\]
2009 VJIMC, Problem 2
Prove that the number
$$2^{2^k-1}-2^k-1$$is composite (not prime) for all positive integers $k>2$.
2024 ELMO Shortlist, G4
In quadrilateral $ABCD$ with incenter $I$, points $W,X,Y,Z$ lie on sides $AB, BC,CD,DA$ with $AZ=AW$, $BW=BX$, $CX=CY$, $DY=DZ$. Define $T=\overline{AC}\cap\overline{BD}$ and $L=\overline{WY}\cap\overline{XZ}$. Let points $O_a,O_b,O_c,O_d$ be such that $\angle O_aZA=\angle O_aWA=90^\circ$ (and cyclic variants), and $G=\overline{O_aO_c}\cap\overline{O_bO_d}$. Prove that $\overline{IL}\parallel\overline{TG}$.
[i]Neal Yan[/i]